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Propagation on a central fiber surrounded by inactive fibers in a multifibered bundle model. - Annals of biomedical engineering

We studied uniform propagation on a central active fiber surrounded by inactive fibers in a multifibered bundle model lying in a large volume conductor. The behavior of a fully active bundle is considered in a companion paper. The bundle is formed by concentric layers of small cylindrical fibers (radius 5 microns), with a uniform minimum distance (d) between any two adjacent fibers, to yield a bundle radius of about 72 microns. Individual fibers are identical continuous cables of excitable membrane based on a modified Beeler-Reuter model. The intracellular volume fraction (fi) increases to a maximum of about 90% as d is reduced and remains unchanged for d < 0.01 micron. In the range of d < 0.01 micron, the central fiber is effectively shielded from external effects by the first concentric layer of inactive fibers, and a large capacitive load current flows across the surrounding inactive membranes. In addition, the fiber proximity produces a circumferentially nonuniform current density (proximity effect) that is equivalent to an increased average longitudinal interstitial resistance. The conduction velocity is reduced as d becomes smaller in the range of d < 0.1 micron, the interstitial potential becomes larger, and both the maximum rate of rise and time constant of the foot of the upstroke are increased. On the other hand, for d > 0.1 micron, there are negligible changes in the shape of the upstroke, and the behavior of the central fiber is close to that of a uniform cable in a restricted volume conductor. For d larger than about 1.2 microns, the active fiber environment is close to an unbounded isotropic volume conductor.

Simulation of two-dimensional anisotropic cardiac reentry: effects of the wavelength on the reentry characteristics. - Annals of biomedical engineering

A two-dimensional sheet model was used to study the dynamics of reentry around a zone of functional block. The sheet is a set of parallel, continuous, and uniform cables, transversely interconnected by a brick-wall arrangement of fixed resistors. In accord with experimental observations on cardiac tissue, longitudinal propagation is continuous, whereas transverse propagation exhibits discontinuous features. The width and length of the sheet are 1.5 and 5 cm, respectively, and the anisotropy ratio is fixed at approximately 4:1. The membrane model is a modified Beeler-Reuter formulation incorporating faster sodium current dynamics. We fixed the basic wavelength and action potential duration of the propagating impulse by dividing the time constants of the secondary inward current by an integer K. Reentry was initiated by a standard cross-shock protocol, and the rotating activity appeared as curling patterns around the point of junction (the q-point) of the activation (A) and recovery (R) fronts. The curling R front always precedes the A front and is separated from it by the excitable gap. In addition, the R front is occasionally shifted abruptly through a merging with a slow-moving triggered secondary recovery front that is dissociated from the A front and q-point. Sustained irregular reentry associated with substantial excitable gap variations was simulated with short wavelengths (K = 8 and K = 4). Unsustained reentry was obtained with a longer wavelength (K = 2), leading to a breakup of the q-point locus and the triggering of new activation fronts.

Point source extracellular stimulation of a myocyte model was used to study the efficacy of excitation of cardiac cells, taking into account the shape of the pulse stimulus and its time of application in the cardiac cycle. The myocyte was modeled as a small cylinder of membrane (10 microns in diameter and 100 microns in length) capped at both ends and placed in an unbounded volume conductor. A Beeler-Reuter model modified for the Na+ dynamics served to simulate the membrane ionic current. The stimulus source was located on the cylinder axis, close to the myocyte (50 microns) in order to generate a nonlinear extracellular field (phi e). The low membrane impedance associated with the high frequency component of the make and break of the rectangular current pulse leads to a current flow across the membrane and an abrupt change in intracellular potential (phi i). Because the intracellular space is very small, phi i is nearly uniform over the length of the myocyte and the membrane potential (V = phi i-phi e) is governed by the applied field phi e. There is then a longitudinal gradient of membrane polarization which is the inverse of the gradient of extracellular potential. With an anodal (positive) pulse, for instance, the proximal portion of the myocyte is hyperpolarized and the distal portion is depolarized. Based on this principle and considering the voltage-dependent activation/inactivation dynamics of the membrane, it is shown that a cathodal (negative) pulse is the most efficacious stimulus at diastolic potentials, an anodal current is preferable during the plateau phase of the action potential, and a biphasic pulse is optimal during the relative refractory phase. Thus a biphasic pulse would constitute the best choice for maximum efficacy at all phases of the action potential.

A model study of electric field interactions between cardiac myocytes. - IEEE transactions on bio-medical engineering

The transmission of excitation via electric field coupling was studied in a model comprising two myocytes abutted end-to-end and placed in an unbounded volume conductor. Each myocyte was modeled as a small cylinder of membrane (10 microns in diameter and 100 microns in length) capped at both ends. A Beeler-Reuter model modified for the Na+ current dynamics served to simulate the membrane ionic current. There was no resistive coupling between the myocytes and the intercellular junction consisted of closely apposed pre- and post-junctional membranes, separated by a uniform cleft distance. The membrane current crossing the prejunctional membrane during the action potential upstroke tends to flow out of the cleft, but it is partly prevented from doing so by the shunt resistance constituted by the cleft volume conductor. The prejunctional upstroke gives rise to a pulse of positive potential within the cleft which induces a small capacitive current across the post-junctional membrane to yield a small positive change in the intracellular potential in the post-junctional cell. The net result is an hyperpolarization of the post-junctional cleft membrane and a slight depolarization of the rest of the cell membrane since the extracellular potential outside of the cell is zero. The magnitude of this depolarization is quite small for a flat junctional membrane and it can be increased by membrane folding and interdigitation, so as to increase the junctional membrane area by a factor of 10 or more. Even then the post-junctional depolarization does not reach threshold when the extracellular potential around the post-junctional cell is effectively zero. Threshold depolarization occurs in the presence of a large decrease of post-junctional load, by increasing the junctional membrane capacitance and/or decreasing the volume of the post-junctional cell. Assuming that the normal resistive coupling between two cardiac myocytes is 1-4 M omega, our model study indicates that electric field coupling would then be about two orders of magnitude smaller. However, substantial enhancement of the efficacy of electric field transmission was observed in the case of cells with substantial junctional membrane folding.

A thin sheet of cardiac tissue was modeled as a set of resistively coupled excitable cables with membrane dynamics described by the modified Beeler Reuter model. Transverse connections have a resistance Rn and are regularly distributed with a spacing delta on any given cable, to provide alternating input and output junctions. Flat wave longitudinal propagation corresponds to propagation along a single continuous cable since all units of the network are functionally isolated due to the absence of transverse current flow. Events on a given cable during flat transverse propagation include electrotonic spread of potential from input to output junctions, action potential initiation at input junctions, and collision at output junctions. The propagating two-dimensional transverse wavefront is an undulating transmembrane potential surface with highs at the input junctions and lows at the output junctions. The action potential upstroke is also modulated in a periodic manner with minimum and maximum Vmax at the input and output junctions respectively. Thus, the network is capable of a diversity of dynamic behavior spatially distributed in relation to the specific pattern of transverse connections chosen. Overall, the behavior of the network model is in good agreement with available structural and electrophysiological data on myocardium. In addition, this network topology allows to handle more easily parameters governing propagation and to avoid very large matrices which are costly in computational effort and overall computer time.

Propagation of an elliptic excitation wave front was studied in a two-dimensional model of a thin sheet of cardiac muscle. The sheet model of 2.5 x 10 mm consisted of a set of 100 parallel cables coupled through a regular array of identical transverse resistors. The membrane dynamics was represented by a modified Beeler-Reuter model. We defined the charging factor (CF) to represent by a single number the proportion of input current used to charge the membrane locally below threshold and showed that CF is inversely correlated with the time constant of the foot of the action potential (tau foot) during propagation on a cable. A safety factor of propagation (SF) was also defined for the upstroke of the action potential, with SF directly correlated with the maximum rate of depolarization (Vmax) and, for cablelike propagation, with propagation velocity. Propagation along the principal longitudinal axis of the elliptic wave front is cablelike but, in comparison with a flat wave front, transverse current flow provides a drag effect that somewhat reduces the propagation velocity, Vmax, SF, and CF. With a longitudinal-to-transverse velocity ratio of 3:1 or more, the wave front propagating along the principal transverse axis is essentially flat and is characterized by multiple collisions between successive pairs of input junctions on a given cable; Vmax, SF, and CF are larger than for longitudinal propagation, but CF is no longer correlated with tau foot. There are transient increases in propagation velocity and Vmax with distance from the stimulation site along both principal axes until stablized values are achieved, and a similar transient decrease in tau foot. Away from the principal axes, the action potential characteristics change progressively along the elliptic wave front.(ABSTRACT TRUNCATED AT 250 WORDS)

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